3 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 3 S2. PROOF OF LEMMA 6 The idea of the proof is as follows. We construct a Taylor expansion to compute the utility gains for any trade. Then we define the traded amount ζ and the utility gain Δ satisfying 9) and 10). Choose any two portfolios x A x B X and any two beliefs δ A δ B [0 1] such that Mx B δ B ) Mx A δ A )>ε. Pick a price p sufficiently close to the middle of the interval between the marginal rates of substitution: p [ Mx A δ A ) + ε/2 Mx B δ B ) ε/2 ] This price is chosen so that both agents will make positive gains. Consider agent A and a traded amount ζ ζ for some ζ which we will properly choose below). The current utility gain associated with the trade z = ζ p ζ) can be written as a Taylor expansion: 51) Ux A z δ A ) Ux A δ A ) = πδ A )u x 1 A) ζ + 1 πδ A ) ) u x 2 A) p ζ + 1 πδa )u y 1) + 1 πδ A ) ) u y 2) p 2) ζ πδ A ) ) u x 2 A) ε/2) ζ + 1 2[ πδa )u y 1) + 1 πδ A ) ) u y 2) p 2] ζ 2 for some y 1 y 2 ) [x 1 A x1 A ζ] [x 2 + p ζ x2 A A ]. The inequality above follows because p Mx A δ A ) + ε/2. An analogous expansion can be done for agent B. Now we want to bound the last line in 51). To do so, we first define the minimal and the maximal prices for agents with any belief in [0 1] and any portfolio in X: { } p = min Mx δ) + ε/2 x X δ [0 1] { } p = max Mx δ) ε/2 x X δ [0 1] These prices are well defined, as X is a compact subset of R 2 ++ and u ) has continuous first derivative on R 2. Then, choose ++ ζ >0 such that, for all ζ ζ and all p [p p], the trade z = ζ p ζ) satisfies z <θand x + z and x z are in R 2 + for all x X. This means that the trade is small enough. Next, we separately bound from below the two terms in the last line of the Taylor

8 8 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI Now we provide an important concept. We want to focus on the utility gains that can be achieved by small trades of norm less than θ), by agents with marginal rates of substitution sufficiently different from each other by at least ε/2). Formally, we proceed as follows. Take any θ>0. Using Lemma 6, we can then find a lower bound for the utility gain Δ>0 from trade between two agents with marginal rates of substitution differing by at least ε/2 and with portfolios in X, making trades of norm less than θ. It is important to notice that this is the gain achieved if the agents trade but do not change their beliefs. Therefore, it is also important to bound from below the gains that can be achieved by such trades if beliefs are updated in the most pessimistic way. This bound is also given by Lemma 6, which ensures that λδ is a lower bound for the gains of the agent offering z at any possible ex post belief where λ is a positive scalar independent of θ). Next, we want to restrict attention to agents who are close to their long-run expected utility. Per period utility u t converges to the long-run value ˆv t,by Lemma 2. We can then apply Lemma 4 and find a time period T T such that, for all t T and for all s: 55) and 56) Pu t ˆv t αδ/4 x t X s) > 1 ε/2 P Mx t δ t ) κ t s) < ε/4 ut ˆv t αδ/8 δ t = δ I s) x t X s ) >α/2 Equation 55) states that there are enough agents, both informed and uninformed, close to their long-run utility. Equation 55) states that there are enough informed agents close to both their long-run utility and to their longrun marginal rates of substitution. We are now done with the preliminary steps ensuring proper convergence and can proceed to the body of the argument. Choose any t T. By Proposition 2, two cases are possible: either i) the informed agents long-run marginal rates of substitution are far enough from each other, κ t s 1 ) κ t s 2 ) ε; or ii) they are close to each other, κ t s 1 ) κ t s 2 ) < ε, but there is a large enough mass of uninformed agents with marginal rate of substitution far from that of the informed agents, P Mx t δ t ) κ t s) 2 ε s) ε for all s. In the next two steps, we construct the desired trade z for each of these two cases, and then complete the argument in step 3. Step 1. Consider the first case, in which κ t s 1 ) κ t s 2 ) ε. In this case, an uninformed agent can exploit the difference between the informed agents marginal rates of substitution in states s 1 and s 2,makinganofferatanintermediate price. This offer will be accepted with higher probability in the state

9 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 9 in which the informed agents marginal rate of substitution is higher. In particular, suppose κ t s 2 ) + ε κ t s 1 ) the opposite case is treated symmetrically). Lemma 6 and the definition of the utility gain Δ imply that there is a trade z = ζ pζ),withpricep = κ t s 1 ) + κ t s 2 ))/2 andsize z <θ, that satisfies the following inequalities: 57) 58) Ux t + z δ t ) u t + Δ if Mx t δ t )>κ t s 1 ) ε/4 andx t X Ux t z δ t ) u t + Δ if Mx t δ t )<κ t s 2 ) + ε/4 andx t X Equation 57) states that all informed and uninformed) agents with marginal rate of substitution above κ t s 1 ) ε/4) will receive a utility gain Δ from the trade z, in terms of current utility. Equation 58) states that all informed and uninformed) agents with marginal rate of substitution below κ t s 1 ) + ε/4) will receive a utility gain Δ from the trade z, in terms of current utility. Combining conditions 56)and57) shows that, in state s 1, there are at least α/2 informed agents with after-trade utility above the long-run utility, Ux t + z δ t )>ˆv t. Since all these agents would accept the trade z, this implies that the probability of acceptance of the trade is χ t z s 1 )>α/2. Next, we want to show that the trade z is accepted with sufficiently low probability conditional on s 2. In particular, we want to show that χ t z s 2 )<α/4. The key step here is to make sure that the trade is rejected not only by informed but also by uninformed agents. The argument is that if this trade were to be accepted by uninformed agents, then informed agents should be offering z and gaining in utility. Formally, proceeding by contradiction, suppose that the probability of z being accepted in state s 2 is large: χ t z s 2 ) α/4. Condition 56) implies that there is a positive mass of informed agents with Mx t δ t )<κ t s 2 ) + ε/2, x t X, and close enough to the long-run utility u t ˆv t αδ/8. By 58), these agents would be strictly better off making the offer z and consuming x t z if the offer is accepted and consuming x t if it is rejected, since 1 χt z t s 2 ) ) Ux t δ t ) + χ t z t s 2 )Ux t z δ t )>u t + αδ/4 > ˆv t Since this strategy dominates the equilibrium payoff, this is a contradiction, proving that χ t z s 2 )<α/4. Step 2. Consider the second case, in which the long-run marginal rates of substitution of the informed agents are close to each other and there is a large enough mass of uninformed agents with marginal rate of substitution far from that of the informed agents.

10 10 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI The argument is as follows: with positive probability, we can reach a point where it is possible to separate the marginal rates of substitution of a group of uninformed agents from the marginal rates of substitution of a group of informed agents. This means that the uninformed agents in the first group can make an offer z to the informed agents in the second group and they will accept the offer in both states s 1 and s 2. If the probabilities of acceptance χ t z s 1 ) and χ t z s 2 ) are sufficiently close to each other, this would be a profitable deviation for the uninformed, since their ex post beliefs after the offer is accepted would be close to their ex ante beliefs. In other words, in contrast to the previous case, they would gain utility but not learn from the trade. It follows that the probabilities χ t z s 1 ) and χ t z s 2 ) must be sufficiently different in the two states, which leads to either 4) or to 5). To formalize this argument, consider the expected utility of an uninformed agent with portfolio x t and belief δ t, who offers a trade z and stops trading from then on: u t + δ t χ t z s 1 ) Ux t z 1) Ux t 1) ) + 1 δ t )χ t z s 2 ) Ux t z 0) Ux t 0) ) where u t is the expected utility if the offer is rejected and the following two terms are the expected gains if the offer is accepted, respectively, in states s 1 and s 2. This expected utility can be rewritten as 59) u t + χ t z s 1 ) Ux t z δ t ) Ux t δ t ) ) + 1 δ t ) χ t z s 2 ) χ t z s 1 ) ) Ux t z 0) Ux t 0) ) using the fact that Ux t δ t ) = δ t Ux t 1) + 1 δ t )Ux t 0) by the definition of U). To interpret 59), notice that, if the probability of acceptance was independent of the signal, χ t z s 1 ) = χ t z s 2 ), then the expected gain from making offer z would be equal to the second term: χ t z s 1 )Ux t z δ t ) Ux t δ t )). The third term takes into account that the probability of acceptance may be different in two states, that is, χ t z s 2 ) χ t z s 1 ) may be different from zero. An alternative way of rearranging the same expression yields 60) u t + χ t z s 2 ) Ux t z δ t ) Ux t δ t ) ) + 1 δ t ) χ t z s 1 ) χ t z s 2 ) ) Ux t z 1) Ux 1) ) In the rest of the argument, we will use both 59)and60). Suppose that there exist a trade z and a period t which satisfy the following properties: a) the probability that z is accepted in state 1 is large enough, χ t z s 1 )>α/4

14 14 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI S6. PROOF OF LEMMA 9 Let us begin from the first part of the lemma. Suppose, by contradiction, that κ t+j s 1 ) κ t s 1 ) >ε for some ε>0 for infinitely many periods. Then, at some date t, an informed agent with marginal rate of substitution close to κ t s) can find a profitable deviation by holding on to his portfolio x t for J periods and then trade with other informed agents at t + J. Let us formalize this argument. Suppose, without loss of generality, that κ t+j s 1 )>κ t s 1 ) + ε for infinitely many periods the other case is treated in a symmetric way). Next, using our usual steps and Proposition 1, it is possible to find a compact set X, atimet, and a utility gain Δ>0 such that the following two properties are satisfied: i) in all periods t T, there is at least a measure α/2 of informed agents with marginal rate of substitution sufficiently close to κ t s), utility close to its long-run level, and portfolio x t in X, that is, 69) P Mx t δ t ) κ t s) <ε/3 ut ˆv t γ J αδ/2 x t X δ t = 1 s ) >α/2 and ii) in all periods t T in which κ t+j s) > κ t s) + ε, there is a trade z such that 70) Ux z 1)>Ux 1) + Δ if Mx 1)<κ t s) + ε/3andx X and 71) Ux+ z 1)>Ux 1) + Δ if Mx 1)>κ t+j s) ε/3andx X Pick a time t T in which κ t+j s) > κ t s) + ε and consider the following deviation. Whenever an informed agent reaches time t and his portfolio x t satisfies Mx t 1) <κ t s) + ε/3 andx t X, he stops trading for J periods and then makes an offer z that satisfies 70) and71). If the offer is rejected, he stops trading from then on. The probability that this offer is accepted at time t + J must satisfy χ t+j z s 1 )>α/2, because of conditions 69) and71). Therefore, the expected utility from this strategy, from the point of view of time t,is u t + γ J χ t+j z s 1 ) Ux t z 1) u t ) >ut + γ J αδ/2 ˆv t so this strategy is a profitable deviation and we have a contradiction. The second part of the lemma follows from the first part, using Proposition 1 and the triangle inequality.

15 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 15 S7. EXAMPLES In this section, we present two examples in which the equilibrium can be analyzed analytically. The first objective of these examples is to show existence of equilibrium with symmetry across states and uniform market clearing in some special cases. The second objective is to study how information acquisition takes place in equilibrium. The third objective is to study how the equilibrium is affected by changing the parameter γ, which controls the probability that the game ends. In particular, we are interested in what happens when γ goes to 1. Higher values of γ correspond to economies in which agents have the chance to do more rounds of trading before the game ends. We can interpret the limit γ 1 as a frictionless trading limit, in which agents have the chance to make infinite rounds of trading before the game ends. In a full information economy in which agents can trade forever i.e., with no discounting), Gale 1986a) showed that bilateral bargaining yields a Walrasian outcome, in which agents making the first offer do not have any monopoly power, due to the fact that their partners have unlimited chances to make further trades in the future. We can then ask whether a similar result applies in our model with asymmetric information when γ goes to 1. Our first example shows that, in general, the result does not extend. In particular, in that example, agents remain uninformed even as γ goes to 1. This bounds the gains from trade that can be reaped by responders who refuse to trade in the first round. Therefore, agents who are selected as proposers in the first period keep some monopoly power. In our second example, on the other hand, uninformed agents have the opportunity to acquire perfect information in equilibrium and payoffs converge to those of perfect competition. In all the examples, there are two types, with initial portfolios x 1 0 = ω 1 ω) and x 2 0 = 1 ω ω),forsomeω 1/2 1).Letφs 1 ) = ϕ>1/2 and recall that, by symmetry, φs 2 ) = 1 ϕ. Informed agents with endowment x 1 0 are called rich informed agents in state s 1 and poor informed agents in state s 2, as their endowment s present value is greater in the first case. The opposite labels apply to informed agents with x 2 0. For analytical tractability, we modify the setup of our model in the first round of trading and make the following assumption. In period t = 1, the matching process is such that agents meet other agents with complementary endowments with probability 1, that is, type 1 agents only get matched with type 2 agents. In all following periods, agents meet randomly as in the setup of Section 1. All our results from previous sections hold in this modified environment, as they only rely on the long-run properties of the game. The purpose of this assumption is to construct equilibria in which almost all trades take place in the first round. We consider two examples. In the first example, uninformed agents do not learn anything about the state s and keep their initial beliefs at δ = 1/2. In the second example, all uninformed agents learn the state s exactly in the first round of trading. For ease of exposition, we present the main results for the two

16 16 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI examples in Sections S7.1 and S7.2, and present some more technical derivations behind the examples in Sections S7.3 and S7.4. S7.1. Example 1: An Equilibrium With No Learning For this example, we introduce an additional modification to our baseline model, assuming that, in period t = 1, all informed agents get to be proposers with probability 1. In particular, in t = 1, informed agents are only matched with uninformed agents and the informed agent is always selected as the proposer. 11 If two uninformed agents meet at t = 1, each is selected as the proposer with probability 1/2. From period t = 2 onward, the matching and the selection of the proposer are as in the baseline model. That is, each agent has the same probability of meeting an informed or uninformed partner and each agent has probability 1/2 of being selected as the proposer. As pointed out above, the changes made in period t = 1 do not affect the long-run properties of the game and the general results of the previous sections still hold. For a given scalar η 0 1), to be defined below, our aim is to construct an equilibrium in which strategies and beliefs satisfy: S1. In t = 1, all proposers of type i offer z i E = η η) x i 0. All responders accept. S2. In t = 2 3, all proposers offer zero trade. S3. In t = 2 3, all uninformed responders reject any offer z that satisfies min { ϕz ϕ)z 2 1 ϕ)z 1 + ϕz 2} < 0; all informed responders reject any offer z that satisfies ϕz ϕ)z 2 < 0 if s = s 1 or 1 ϕ)z 1 + ϕz 2 < 0 if s = s 2 B1. Uninformed responders keep their beliefs unchanged after offer z i E in period t = 1 and after offer 0 in period t 2. B2. In t = 1, after an offer z z i E, uninformed responders adjust their belief to δ = 1 if they are of type 1 and to δ = 0 if they are of type 2. B3. In t = 2 3, after an offer z 0, uninformed responders adjust their belief to δ = 1ifϕz ϕ)z 2 <1 ϕ)z 1 + ϕz 2 and to δ = 0ifϕz ϕ)z 2 >1 ϕ)z 1 + ϕz 2. Notice that, in equilibrium, all agents reach endowments on the 45 degree line after one round of trading and remain there from then on. S1 S3 and B1 B3 describe strategies and beliefs in equilibrium and along a subset of off-the-equilibrium-path histories. This is sufficient to show that we have an equilibrium, since we can prove that, if other agents strategies satisfy S1 S3, 11 This requires assuming α<1/2.

17 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 17 the payoff from any deviating strategy is bounded above by the equilibrium payoff. An important element of our construction is that following off-theequilibrium-path offers at t 2, uninformed agents hold pessimistic beliefs, meaning that they expect the state s to be the one for which the present value of the offer received is smaller. This, together with the fact that all agents are on the 45 degree line starting at date 2, implies that deviating agents have limited opportunities to trade after period 1. The argument to prove that an equilibrium with these properties exists is in two steps. First, we show that zero trade from period t = 2 on is a continuation equilibrium. Second, we go back to period t = 1 and show that making and accepting the offers z i E is optimal at t = 1. To show that no trade is an equilibrium after t = 2, we use property S3. Let Vx δ)denote the continuation utility of an agent with endowment x and belief δ [0 1] at any time t This value function is independent of t since the environment is stationary after t = 2. Since other agents strategies satisfy S3, the endowment process of a deviating agent who starts at x δ) satisfies the following property: if the state is s 1, any endowment x reached with positive probability at future dates must satisfy ϕ x ϕ) x 2 ϕx ϕ)x 2 This property holds because, in s 1, neither informed nor uninformed agents will accept trades that increase the expected value of the proposer s endowment computed using the probabilities ϕ and 1 ϕ. A similar property holds in s 2, reversing the roles of the probabilities ϕ and 1 ϕ. These properties, together with concavity of the utility function, imply that the continuation utility Vx δ) is bounded as follows: 72) Vx δ) δu ϕx ϕ)x 2) + 1 δ)u 1 ϕ)x 1 + ϕx 2) In the continuation equilibrium, all agents start from a perfectly diversified endowment with x 1 = x 2. So an agent can achieve the upper bound in 72) by not trading. This rules out any deviation by proposers on the equilibrium path. A similar argument shows that the off-the-equilibrium-path responses in S3 are optimal. 13 So we have an equilibrium for t 2. The argument for no trade in periods t 2 is closely related to the no-trade theorem of Milgrom and Stokey 1982). Turning to period t = 1, consider a rich informed proposer with endowment x 1 0 in state s 1. If he deviates and offers z z 1 E, the uninformed responder s belief goes to δ = 0. Then the offer will only be accepted if Vx z 0) Vx 1 0 0) and the payoff of the proposer, if the offer is accepted, would be 12 The value function is defined at the beginning of the period, before knowing whether the game ends or there is another round of trading. 13 See Proposition 6 in Section S7.3.

18 18 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI Vx 1 0 z 1). In Section S7.3, we derive an upper bound on this payoff. We can then find parametric examples and choose η so that offering z 1 E yields a higher payoff. The reason why this is possible is that, if the proposer offers z 1 E, the uninformed responder s belief remains at δ = 1/2, so the informed proposer is able to trade at better terms in period 1. Consider next a poor informed proposer with endowment x 2 0 in state s 1.If he deviates, the uninformed responder adjusts his belief to δ = 1. Since the poor informed proposer also holds belief δ = 1, we can show that the best deviation by a poor informed proposer is to make an offer that reaches the 45 degree line for both. However, the responder s outside option is higher at δ = 1 than at δ = 1/2, because he holds a larger endowment of asset 1. This makes the participation constraint of the responder tighter than at the equilibrium offer z 2 E and allows us to construct parametric examples in which the poor informed proposer prefers not to deviate. Having shown that both the rich informed proposer and the poor informed proposer prefer not to deviate, we can then show that an uninformed proposer prefers not to deviate either, using the argument that the payoff of an uninformed agent under a deviation is weakly dominated by the average of the payoffs of an informed agent with the same endowment. In Section S7.3, we derive sufficient conditions that rule out deviations in period t = 1 and show how to construct parametric examples that satisfy these conditions. The following proposition contains such an example. PROPOSITION 4: Suppose the utility function is uc) = c 1 σ /1 σ) and the parameters σ ω ϕ) are in a neighborhood of 4 9/ / )). There is an η 0 1) and a cutoff ᾱ>0 for the fraction of informed agents in the game, such that S1 S3 and B1 B3 form an equilibrium if 0 α<ᾱ. An important ingredient in the construction of the example in the proposition is to assume that the fraction of informed agents α is sufficiently small. This puts a bound on the utility from trading in periods t 2, because it implies that an agent only gets a chance to trade with informed agents with a small probability. In particular, property S3 means that an agent who starts at x at t = 2 and only trades with uninformed agents before the end of the game can only reach endowments x that satisfy both and ϕ x ϕ) x 2 ϕx ϕ)x 2 1 ϕ) x 1 + ϕ x 2 1 ϕ)x 1 + ϕx 2 This restriction is crucial in constructing upper bounds on the continuation utility of deviating agents at date t = 1. The intuition is that the trades z i E at date t = 1 are proposed and accepted in equilibrium because the outside

19 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 19 option is to trade with fully diversified, mostly uninformed agents in periods t = 2 3.Inperiodt = 1, there are large gains from trade coming from the fact that agents are not diversified, but all these gains from trade are exhausted in the first round of trading. From then on, the presence of asymmetric information limits the gains from trade for a deviating agent who is still undiversified at the end of t = 1. Somewhat surprisingly, under the assumptions in Proposition 4 an equilibrium can be constructed for any value of γ. This is because what bounds the continuation utility in period t = 2 is the small probability of trading with informed agents in future periods. So for any value of γ, we can choose the cutoff ᾱ sufficiently small to make the probability of trading with informed agents approach zero and sustain our equilibrium. Larger values of γ correspond to economies in which agents trade more frequently before the game ends. Then the observation above can be interpreted as follows. There can be economies close to the frictionless limit that is, with γ close to 1 in which no information is revealed in equilibrium. This happens because more frequent trade implies that the diversification motive for trade is exhausted more quickly. However, once the diversification motive is exhausted, the no-trade theorem implies that no further trade occurs and so no further information is revealed. When the mass of informed agents α is zero, the equilibrium holds for all γ, sowecantakeγ 1. We then have an economy in which the limit equilibrium allocation is an allocation in which uninformed agents with the same endowments get different consumption levels depending on whether they were selected as proposers or responders in the very first period of the game. So we have an example of an economy in which the presence of more frequent rounds of trading does not lead, in the limit, to a perfectly competitive outcome, unlike in the economies with perfect information analyzed by Gale 1986a). This shows that the presence of asymmetric information can have powerful effects in decentralized economies. Again, the underlying idea is that the no-trade theorem limits the agents ability to trade in the long run, and this induces agents to accept trades in the early stages of the game, when diversification motives are stronger. This undermines the ability of future rounds of trading to act as a check on the monopoly power of proposers in the early stages of the game. S7.2. Example 2: An Equilibrium With Learning We now turn to an example in which uninformed agents acquire perfect information in the first round of trading. As in Example 1, we assume that, in the first round of trading, each agent is matched with an agent with complementary endowments. Moreover, we assume that almost all agents are informed, so there is only a zero mass of uninformed agents. We also assume that preferences display constant absolute risk aversion: uc) = e ρa

20 20 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI This assumption allows us to characterize analytically the value function Vx δ)for δ = 0andδ = 1. Notice that CARA preferences do not satisfy the property lim c 0 uc) =, which was assumed in Section 2 Assumption 2). However, the only purpose of that property was to ensure that endowments stay in a compact set with probability close to 1 in equilibrium. Here, we can check directly that endowments remain in a compact set in equilibrium. So all our general results still apply. The analysis of this example proceeds in two steps. First, we characterize the equilibrium focusing on the behavior of informed agents which can be done, given that uninformed agents are in zero mass. Second, we look at the uninformed agent s problem at date t = 1 and derive conditions that ensure that his optimal strategy is to experiment, making an offer that perfectly reveals the state s. For the first step, we need to derive four equilibrium offers z i E s) which depend on the proposer type i and on the state s. In equilibrium, all proposers of type i make offer z i E s) in state s, all responders accept, and both proposer and responder reach a point on the 45 degree line. The offers z i E s) are found maximizing Vx i 0 z δ) subject to Vx i 0 + z δ) Vx i 0 δ) with δ = 1ifs 1 and δ = 0ifs 2. Since the two agents share the same beliefs, it is not difficult to show that the solution to this problem yields an allocation on the 45 degree line for both agents and that they stop trading from period t = 2 onward. For this argument, it is sufficient to use the upper bound 72), which was used for our Example 1 and also holds here. We then obtain the following proposition. 14 PROPOSITION 5: If all agents are informed, there is an equilibrium in which all agents reach the 45 degree line in the first round of trading and stop trading from then on. Before turning to our second step, however, we need to derive explicitly the form of the V function and the offers z i E s). These steps are more technical and are presented in Section S7.4. The assumption of CARA utility helps greatly in these derivations, as it allows us to show that the value function takes the form Vx)= exp{ ρx 1 }fx 2 x 1 ) for some decreasing function f and that the function f can be obtained as the solution of an appropriate functional equation. We can then turn to our second step and consider uninformed agents in period t = 1. The case of uninformed responders is easy. Since they meet informed proposers with probability 1 and these proposers make different offers 14 A detailed proof is in Section S7.4.

21 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 21 in the two states, they acquire perfect information on s and respond like informed agents, accepting the offer and reaching the 45 degree line. The case of uninformed proposers is harder. We want to show that an uninformed proposer with endowment i makes offer z i E s 1 ) if i = 1 and offer z i E s 2 ) if i = 2, where z i E s) are the offers derived above for informed agents. In other words, uninformed proposers mimic the behavior of rich informed proposers. By making these offers, uninformed proposers get to learn exactly the state s, because their offer is accepted with probability 1 in one state and rejected with probability 1 in the other. While this implies that they become informed from period t = 2 onward, it also implies that, with probability 1/2, they do not reach the 45 degree line in the first round of trading. Our characterization of the V function in Section S7.4 allows us to compute their payoff in this case and to characterize their trading in periods t = 2 3. In particular, the optimal behavior of uninformed agents who fail to trade in period t = 1is to make an infinite sequence of trades in all periods t 2 in which they are selected as proposers. The difference x 1 x 2 is reduced by a factor of 1/2each time they get to trade, so that they asymptotically reach the 45 degree line. Let us show that the offers described above are optimal for the uninformed proposer. We focus on type i = 1, as the case of i = 2 is symmetric. To prove that offering z 1 E s 1 ) is optimal at time t = 1, we need to check that: 1. offer z 1 E s 1 ) is rejected by informed agents in s 2 ; 2. offer z 1 E s 1 ) dominates any other offer accepted by informed agents only in s 1 ; 3. offer z 1 E s 1 ) dominates any offer accepted by informed agents only in state s 2 ; 4. offer z 1 E s 1 ) dominates any offer accepted by informed agents in both states. Conditions 1 to 3 are proved in Section S7.4 andholdforanychoiceofparameters. The most interesting condition is the last one, which ensures that the uninformed agent prefers to learn even though it entails not trading with probability 1/2 in the first period. In Section S7.4, we derive an upper bound for the expected utility from any offer accepted by informed responders in both s 1 and s 2. Making the following assumptions on parameters: ρ = 1 ϕ= 2/3 ω= 1 we can compute the expected utility from offering z 1 E s 1 ) and the upper bound just discussed. The values we obtain are plotted in Figure S1 for different values of γ. As the figure shows, there is a range of γ for which experimenting dominates non-experimenting, so uninformed agents offer z 1 E s 1 ). Notice that as γ goes to 1, the expected utility of the uninformed agent converges to the expected utility of the informed agent and both converge to the expected utility in a Walrasian rational expectations equilibrium. So unlike in Example 1, in this

22 22 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI FIGURE S1. Example 2. case, as the frequency of trading increases, the equilibrium payoffs converge to those of a perfectly competitive rational expectations equilibrium. As a final remark, notice that, in this example, some agents the uninformed proposers whose offer is rejected in the first round only reach an efficient allocation asymptotically. However, we can characterize the speed at which this convergence occurs. To measure distance from efficiency, let us use the distance d t x 1 t x 2 t. Since this distance is reduced by 1/2 every time the agent gets to make an offer and trade, after n trades in which the agent is selected as the proposer, the distance is reduced to 2 n d 0. It is then possible to show that, for every ɛ, there is a γ large enough that the probability of d t <ɛis largerthan 1 ɛ. That is, with γ close to 1, the allocation approaches efficiency also for uninformed first-round proposers. S7.3. Example 1: Proofs PROPOSITION 6: In the economy of Example 1, individual strategies are optimal for t 2. PROOF: Consider first a proposer, informed or uninformed, who has not deviated up to time t, so he holds an endowment x on the 45 degree line, either η η) or 1 η 1 η),andbeliefsδ [0 1].Frominequality72), the expected payoff from any deviating strategy is bounded above by δu ϕx ϕ)x 2) + 1 δ)u 1 ϕ)x 1 + ϕx 2) = u x 1)

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